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Notes on Factor Substitution With Diagrams

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Peter Tomlinson

Urban Land Markets With Factor Substitution April 2011 In what follows here, we will elaborate on the analysis of land-market equilibrium with factor substitution, as outlined in OSullivans text. Starting with a model in Chapter 6 of the text, we derive the models endogenous variables from its exogenous variables. Factor substitution refers to firms ability to substitute capital for land at locations where land rent is relatively high rather than using fixed land and capital inputs regardless of land rent. The factor-substitution model in the text considers firms that occupy office buildings. However, a factor substitution model can apply to any type of firm. In these notes, factor substitution can be switched on or off so that its impact on endogenous variables can be isolated. A zoning law controlling firms land input is the tool used to switch factor substitution on or off. Firms in the textbook model sell information products to their customers. Customers might be paying for financial advice or legal advice two examples of information products. Firms need data such as data on financial markets or data on matters involved in contract law in order to develop saleable information products. The location variable in the model is distance from the office land markets median location (the location at the centre of a circular land market). We assume that the cost of obtaining required data increases with distance from the median location. Required data is obtained at meetings, and travel cost (an exogenous variable) measures the cost of attending these meetings. We begin with Table 6-7 in the text. 1 We will re-order the table so that exogenous variables (given from outside the model) are grouped in columns 1-4; the endogenous variables (determined in the model) will be grouped in columns 5-9. The resulting table appears here as Table 1. Table 1 includes values for variables on rows omitted in Table 6-7: the rows added here show all data for locations at 0, 4 and 6 blocks from the median location, while travel cost numbers have been added for locations at 2 and 3 1 In what follows here, hyphenated numbers for tablesthr fithres (for example Table 6-7) refer to tables or numbers without a hyphen (for example Table 1) refer to tables or figures included in these notes. Thesee notes cannot be read without the text book at hand, since there will be numerous references to tables and diagrams in the text that are not reproduced in the notes. blocks. The blank spaces for variables at 2 and 3 blocks are not filled in since data at these locations will not be needed in the discussion below. Table 1: Expanded / Re-ordered Version of Table 6-7 in Text 1.Distance 2.Travel 3.Total 4.Other 5.Productn 6.Bldg. 7.Total 8.Capital 9.Bid in blocks cost revenue non-land site height rent paid cost of rent per ( x ) cost (hectares) (floors) bldg. ha 0 $0 $500 $150 0.02 50 $70 $280 $3500 1 $36 $500 $150 0.04 25 $64 $250 $1600 2 $74 $500 $150 3 $114 $500 $150 4 $156 $500 $150 0.125 8 $54 $140 $432 5 $200 $500 $150 0.25 4 $50 $100 $200 6 $246 $500 $150 0.50 2 $29 $75 $58 The numbers in Table 1 could be interpreted as applying to seven firms at the seven locations indicated in Column 1. Alternatively, we could interpret Table 1 as applying to a single firm that moves from one location to another. Either way, the firm or firms will compete with large numbers of identical firms. The conditions for perfect competition are assumed to be met here, including zero economic profit in equilibrium (firms make only the profit required to keep them in business). The profit required to stay in business is included in the other non- land cost variable. Exogenous Variables Column 1 in Table 1 shows distance from the median location, measured in blocks. A block is treated as a standard distance measure, comparable to kilometres or miles. This variable measures distance from the median location in any direction whether east, south, west, north (or any other compass direction) makes no difference. All variables in the model will have equal values at any given x, regardless of direction. Travel cost is shown in Column 2. Since travel cost is an exogenous variable, any numbers could be assumed for Column 2. The assumption in the text is that travel cost increases at an increasing rate moving away from the median location. In the example of Table 1, a move from 0 blocks (the median location itself) to one block away increases travel cost by $36 - $0 = $36 per day; a move from 1 block to 2 blocks increases travel cost by $74 - $36 = $38 per day, and so on. The text provides some of the numbers for travel cost, and in Table 1 numbers have been filled in for other locations. All variables with a time dimension (Columns 2, 3, 4, 7, 8, 9) are specified on a per-day basis. The text indicates that an office firms revenue ($500) is per day, so all other time-specific variables must be measured on a per-day basis. (The time period could alternatively be a month, a year or any other time interval, as long as it is specified consistently for all variables.) 2 The product that is sold for $500 per day is not identified in the text, but identifying it is unnecessary. Whatever the product is, both the output level produced by a firm and the output price are exogenous; price times quantity of output is fixed at $500 per day per firm. Not shown in Table 1 is another exogenous variable assumed to apply equally to each firm. Each firm is assumed to require an office building with 10,000 sq. metres (1 hectare) of floor area to produce its output. The firm could rent that floor area from another firm producing office space for rent. However, in the model here and in the text the firm constructs its own office building, renting land and capital for that purpose. It then uses its office building to produce the information product that sells for $500 per day. Column 4 (other non-land cost) shows a firms cost of inputs other than travel cost, capital cost and land cost. ONLC includes labour cost and the profit required by the firm to stay in business. The assumption is that inputs included in ONLC cannot be substituted for land or capital: their cost is a constant $150 / day given exogenously. Endogenous Variables Columns 5 and 6 in Table 1 are effectively the same variable. Column 5 shows the land input in hectares rented by a firm at each location (x-value), and Column 6 shows that same land input divided into the firms fixed floor area (1 hectare). Thus Column 6, the ratio of floor area to land area, is the reciprocal of Column 5. For example firms at x = 5 rent 0.25 hectares of land on which they construct 1 hectare of floor area, so the floor area / land area ratio is 1 / 0.25 = 4. In these notes we will refer to land input rather the equivalent term production site used in the text. We assume that a firms land input is limited to land under the building (no additional land for landscaping, parking, etc.). We also assume that every floor has the same area as the ground floor, so a building on 0.25 hectares of land has four floors of 0.25 hectares each. Given these assumptions, the floor3area / land area ratio is also building height measured in numbers of floors. 2Other non-land cost may also include travel cost at x = 0. In Table 1 and in the text, travel cost at x = 0 is set equal to zero. However, that is consistent with firms at x = 0 actually incurring travel cost. To achieve that consistency, define travel cost as the difference between travel cost at each x and travel cost at x = 0. Travel cost at x = 0 is then included with the other constants in ONLC.. 3Note: building height is equivalent to the floor area / land area ratio only if that ratio 1.0. The minimum building height is obviously one floor. For example with a floor area / land area ratio = 0.5 (1 hectare floor area on 2 hectares of land), we would either have a building one floor high covering half the site or some taller building covering less than half the site. A building 0.5 floors high covering the entire site is not an option. 3
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